2503prm - Metamath Proof Explorer

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Theorem 2503prm 15847

Description: 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

**Hypothesis**
Ref	Expression
2503prm.1	⊢ 𝑁 = ;;;2503

**Assertion**
Ref	Expression
2503prm	⊢ 𝑁 ∈ ℙ

**Proof of Theorem 2503prm**
Step	Hyp	Ref	Expression
1		139prm 15831	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 11308	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 11191	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 11518	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 11309	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 11312	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 11512	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 11307	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 11512	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 11151	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2622	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 11535	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2647	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 6660	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 11315	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 11512	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 11310	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 11512	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 11316	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2622	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 11313	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 11512	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2622	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2622	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 11314	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2622	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 11102	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 9994	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 11156	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 10228	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 11522	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2644	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 10042	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 10224	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 6662	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 11134	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2644	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 11106	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 10042	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 6660	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 11617	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2644	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 11566	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 11522	. . . . . . 7 ⊢ 6 = ;06
46		3cn 11095	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 10043	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 10224	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 6662	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 11161	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2644	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 11311	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 11655	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 11098	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 11598	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 10228	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 11581	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 11566	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 11568	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 11108	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 10043	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 6660	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 11625	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2644	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 11669	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 10047	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 11587	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 11589	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 11512	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 11304	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 10297	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2647	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2647	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 11512	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2697	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 11304	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 10290	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 708	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2631	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 11031	. 2 ⊢ 1 ∈ ℕ
81		2nn 11185	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 11512	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 15781	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 6661	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2647	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 11674	. . . 4 ⊢ 8 < ;10
87		1lt10 11681	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 11546	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 11532	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 4680	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 15845	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 15846	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 15611	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 − cmin 10266 2c2 11070 3c3 11071 4c4 11072 5c5 11073 6c6 11074 7c7 11075 8c8 11076 9c9 11077 ℕ₀cn0 11292 cdc 11493 ↑cexp 12860 ℙcprime 15385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-odz 15470 df-phi 15471 df-pc 15542

This theorem is referenced by: (None)

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